MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vccl Structured version   Visualization version   GIF version

Theorem vccl 30367
Description: Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vciOLD.1 𝐺 = (1st𝑊)
vciOLD.2 𝑆 = (2nd𝑊)
vciOLD.3 𝑋 = ran 𝐺
Assertion
Ref Expression
vccl ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Proof of Theorem vccl
StepHypRef Expression
1 vciOLD.1 . . 3 𝐺 = (1st𝑊)
2 vciOLD.2 . . 3 𝑆 = (2nd𝑊)
3 vciOLD.3 . . 3 𝑋 = ran 𝐺
41, 2, 3vcsm 30366 . 2 (𝑊 ∈ CVecOLD𝑆:(ℂ × 𝑋)⟶𝑋)
5 fovcdm 7586 . 2 ((𝑆:(ℂ × 𝑋)⟶𝑋𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
64, 5syl3an1 1161 1 ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099   × cxp 5671  ran crn 5674  wf 6539  cfv 6543  (class class class)co 7415  1st c1st 7986  2nd c2nd 7987  cc 11131  CVecOLDcvc 30362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-1st 7988  df-2nd 7989  df-vc 30363
This theorem is referenced by:  vc0  30378  vcm  30380  nvscl  30430
  Copyright terms: Public domain W3C validator